# MATH 6710: Probability Theory I

## Fall 2016

### Basic information:

Meeting time: MWF 12:20-1:10 pm
Location: Rockefeller Hall 132
Instructor: Daniel Jerison
Office: Malott Hall 581
Office hours: Tu 2-3 pm, Th 3-4 pm or by appointment
Email: jerison at math.cornell.edu

TA: Vardan Verdiyan
Office hours: W 3-4 pm, Th 2-3 pm or by appointment, 657 Rhodes Hall, Conference Room 1
Email: vv98 at cornell.edu

### Course description:

A mathematically rigorous course in probability theory which uses measure theory but begins with the basic definitions of independence and expected value in that context. Law of large numbers, Poisson and central limit theorems, and random walks.

Prerequisites: Knowledge of Lebesgue integration theory, at least on real line. Students can learn this material by taking parts of MATH 4130-4140 or MATH 6210.

### Textbook:

Probability: Theory and Examples, 4th edition, by Rick Durrett. We will cover most of Chapters 1-4.

Other textbooks that you can look at for a different perspective:
Measure Theory by Cohn or Real Analysis: Modern Techniques and Their Applications by Folland for measure theory background.
Probability and Measure by Billingsley, A Course in Probability Theory by Chung, A First Look at Rigorous Probability Theory by Rosenthal for the main subject material of the course.

### Lecture notes:

These notes were written by John Pike for last year's version of this class. Here are the full notes for the entire semester (document is searchable and has working links). Following are links to the individual sections we have covered so far:

Section 1: Introduction

Section 2: Preliminary results

Section 3: Distributions

Section 4: Random variables

Section 5: Expectation

Section 6: Independence

Section 7: Weak law of large numbers

Section 8: Borel-Cantelli lemmas

Section 9: Strong law of large numbers

Section 10: Random series

Section 11: Weak convergence (with supplement 1 and supplement 2)

Section 12: Characteristic functions

Section 13: Central limit theorems

Section 14: Poisson convergence

Section 15: Stein's method (we are skipping this section)

Section 16: Random walk preliminaries

Section 17: Stopping times

Section 18: Recurrence

Section 19: Path properties

Section 20: Law of the iterated logarithm

Through Friday, December 2, we have gotten through the end of Section 19 and the statement of the Law of the Iterated Logarithm in Section 20.

Your grade will be determined by weekly homework assignments along with one take-home final exam. The approximate weighting will be 70% homework and 30% final.

### Final exam:

Here is the take-home final exam. Solutions. It is due on Thursday, December 8 at 11:30 AM. You may consult any printed or online source, but you must explicitly cite all sources besides the textbook and lecture notes. Apart from asking me to clarify the questions, you may not get help from any person on the take-home final. In particular, you are not allowed to work with each other. The exam should be submitted via email or to my office, slid under the door if I am not there.

### Homework:

Homework will usually be due on Fridays. Each student is granted two free passes to turn in homework up to a week after the posted due date. Beyond this, late work will not be accepted without a compelling reason. You may not use a late pass on the final homework assignment.

You are encouraged to work with each other on the homework. Everything you write should be in your own individual words; direct copying is forbidden! Also, though it is perfectly acceptable to consult outside resources for help on occasion, you should spend a reasonable amount of time thinking about the problem and attempting your own solution before doing so, and you are required to explicitly cite all sources other than the official text.

HW 1: Due Friday, September 2. Solutions.

HW 2: Due Friday, September 9. Solutions.

HW 3: Due Friday, September 16. Solutions.

HW 4: Due Friday, September 23. Solutions.

HW 5: Due Friday, September 30. Solutions.

HW 6: Due Friday, October 7. Solutions.

HW 7: Due Friday, October 14. Solutions.

HW 8: Due Friday, October 21. Solutions and Solution to extra credit problem.

HW 9: Due Friday, October 28. Solutions.

HW 10: Due Friday, November 4. Solutions.

HW 11: Due Friday, November 11. Solutions.

HW 12: Due Friday, November 18. Solutions.

HW 13: Due Wednesday, November 30. Solutions.

### Disabilities:

If you have a disability-related need for reasonable academic adjustments in this course, please provide me with an accommodation notification letter from Student Disability Services.